egbToric -- computes the kernel of an equivariant monomial map

Usage:

G = egbToric m
Inputs:
- m, a ring map, an equivariant monomial map between rings with symmetric action
Optional inputs:
- OutFile => ..., default value null, where to send messages
Outputs:
- G, a list, an equivariant Gröbner basis for the kernel of the map

Description

m should be a monomial map between rings created by buildERing. Such a map can be constructed with buildEMonomialMap but this is not required.

For a map to ring R from ring S, the algorithm infers the entire equivariant map from where m sends the variable orbit generators of S. In particular for each orbit of variables of the form x_{(i_1,...,i_k)}, the image of x_{(0,...,k-1)} is used.

egbToric uses an incremental strategy, computing Gröbner bases for truncations using FourTiTwo. Because of FourTiTwo's efficiency, this strategy tends to be much faster than general equivariant Gröbner basis algorithms such as egb.

In the following example we compute an equivariant Gröbner basis for the vanishing equations of the second Veronese of P^n, i.e. the variety of n x n rank 1 symmetric matrices.

i1 : R = buildERing({symbol x}, {1}, QQ, 2);

i2 : S = buildERing({symbol y}, {2}, QQ, 2);

i3 : m = buildEMonomialMap(R,S,{x_0*x_1})

                  2               2
o3 = map (R, S, {x , x x , x x , x })
                  1   1 0   1 0   0

o3 : RingMap R <-- S

i4 : G = egbToric(m, OutFile=>stdio)
3
     -- used .00208812 seconds
     -- used .00056512 seconds
(9, 9)
new stuff found
4
     -- used .00381056 seconds
     -- used .00465036 seconds
(16, 26)
new stuff found
5
     -- used .00845432 seconds
     -- used .0277724 seconds
(25, 60)
6
     -- used .019187 seconds
     -- used .204988 seconds
(36, 120)
7
     -- used .04195 seconds
     -- used .854451 seconds
(49, 217)

                                   2
o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
         1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
     ------------------------------------------------------------------------
     y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
      2,0 1,1     2,2 1,0    2,1 2,0     3,2 1,0    3,0 2,1     3,2 1,0  
     ------------------------------------------------------------------------
     y   y   }
      3,1 2,0

o4 : List

Caveat

It is not checked if m is equivariant. Only the images of the orbit generators of the source ring are examined and the rest of the map ignored.

Ways to use egbToric:

egbToric(RingMap)

For the programmer

The object egbToric is a method function with options.

The source of this document is in EquivariantGB.m2:1194:0.

egbToric -- computes the kernel of an equivariant monomial map

Description

Caveat

See also

Ways to use egbToric:

For the programmer